# Generic minimizing behavior in semi-algebraic optimization

arXiv: https://arxiv.org/abs/1504.07694 · HTML: https://mlsanity.com/paper/1504.07694/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2015.

## Built on by (most cited first, 5 of 5)

- [Asymptotic normality and optimality in nonsmooth stochastic approximation](https://mlsanity.com/paper/2301.06632/) (2023)
- [Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization](https://mlsanity.com/paper/2108.11832/) (2021)
- [Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions](https://mlsanity.com/paper/2108.02072/) (2021)
- [Escaping strict saddle points of the Moreau envelope in nonsmooth optimization](https://mlsanity.com/paper/2106.09815/) (2021)
- [Stochastic Subgradient Descent on a Generic Definable Function Converges to a Minimizer](https://mlsanity.com/paper/2109.02455/) (2021)

## Abstract (arXiv)

We present a theorem of Sard type for semi-algebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, for a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semi-algebraic optimization problems: "typical" semi-algebraic problems have finitely many critical points, around each of which they admit a unique "active manifold" (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary.